Cartan–Karlhede algorithm: Difference between revisions

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The '''Cartan–Karlhede algorithm''' is a procedure for completely classifying and comparing [[Riemannian manifold]]s. Given two [[Riemannian manifold]]s of the same dimension, it is not always obvious whether they are [[local isometry|locally isometric]].<ref>{{cite book | author=Olver, Peter J. |author-link=Peter J. Olver | title=Equivalents, Invariants, and Symmetry | ___location=Cambridge | publisher=[[Cambridge University Press]] | year=1995 | isbn=0-521-47811-1}}</ref> [[Élie Cartan]], using his [[exterior derivative|exterior calculus]] with his method of [[moving frames]], showed that it is always possible to compare the manifolds. [[Carl Brans]] developed the method further,<ref>{{citation|first1=Carl H.|last1=Brans|title=Invariant Approach to the Geometry of Spaces in General Relativity|journal=J. Math. Phys.|volume=6|page=94|year=1965|doi=10.1063/1.1704268|bibcode = 1965JMP.....6...94B }}</ref> and the first practical implementation was presented by {{ill|Anders Karlhede|sv}} in 1980.<ref>{{citation|first1=A.|last1=Karlhede|title=A review of the geometrical equivalence of metrics in general relativity|journal=General Relativity and Gravitation|volume=12|page=693|year=1980|issue=9 |doi=10.1007/BF00771861|bibcode = 1980GReGr..12..693K |s2cid=120666569 }}</ref>

One of the most fundamental problems of [[Riemannian geometry]] is this: given two [[Riemannian manifold]]s of the same dimension, how can one tell if they are [[local isometry|locally isometric]]? This question was addressed by [[Elwin Christoffel]], and completely solved by [[Élie Cartan]] using his [[exterior derivative|exterior calculus]] with his method of [[moving frames]].

The main strategy of the algorithm is to take [[covariant derivative]]s of the [[Riemann tensor]]. Cartan showed that in ''n'' dimensions at most ''n''(''n''+1)/2 differentiations suffice. If the Riemann tensor and its derivatives of the one manifold are algebraically compatible with the other, then the two manifolds are isometric. The Cartan–Karlhede algorithm therefore acts as a kind of generalization of the [[Petrov classification]].

Cartan's method was adapted and improved for [[general relativity]] by A. Karlhede, who gave the first algorithmic description of what is now called the '''Cartan-Karlhede algorithm'''. The algorithm was soon implemented by J. Åman in an early symbolic computation engine, [[SHEEP (symbolic computation system)]], but the size of the computations proved too challenging for early computer systems to handle.

==Physical Applications==

The potentially large number of derivatives can be computationally prohibitive. The algorithm was implemented in an early symbolic computation engine, [[SHEEP (symbolic computation system)|SHEEP]], but the size of the computations proved too challenging for early computer systems to handle.<ref>{{citation|first1=J. E.|last1=Åman|first2=A.|last2=Karlhede|title=A computer-aided complete classification of geometries in general relativity. First results|journal=Phys. Lett. A|volume=80|page=229|year=1980|issue=4 |doi=10.1016/0375-9601(80)90007-9|bibcode = 1980PhLA...80..229A }}</ref><ref>{{citation|first1=J. E.|last1=Åman|title=Manual for CLASSI: classification programs in general relativity|publisher=University of Stockholm Institute of Theoretical Physics}}</ref> For most problems considered, far fewer derivatives than the maximum are actually required, and the algorithm is more manageable on modern computers. On the other hand, no publicly available version exists in more modern software.<ref>{{cite journal |author1=Pollney, D. |author2=Skea, J. F. |author3=d'Inverno, Ray | title=Classifying geometries in general relativity (three parts) | journal=Class. Quantum Grav. | year=2000 | volume=17 |issue=3 | pages=643–663, 2267–2280, 2885–2902 | doi=10.1088/0264-9381/17/3/306|bibcode = 2000CQGra..17..643P |s2cid=250907225 }}</ref>

The Cartan-Karlhede algorithm has important applications in general relativity. One reason for this is that the simpler notion of [[curvature invariant]]s fails to distinguish spacetimes as well as they distinguish [[Riemannian manifold]]s. This difference in behavior is due ultimately to the fact that spacetimes have isotropy subgroups which are subgroups of the [[Lorentz group]] SO⁺(3,'''R'''), which is a ''noncompact'' [[Lie group]], while four-dimensional Riemannian manifolds (i.e., with [[positive definite]] [[metric tensor]]), have isotropy groups which are subgroups of the [[compact]] Lie group SO(4).

==Physical Applicationsapplications==

Cartan's method was adapted and improved for general relativity by A. Karlhede, and

implemented by J. Åman in an early symbolic computation engine, [[SHEEP (symbolic computation system)]].

The Cartan-KarlhedeCartan–Karlhede algorithm has important applications in [[general relativity]]. One reason for this is that the simpler notion of [[curvature invariant]]s fails to distinguish spacetimes as well as they distinguish [[Riemannian manifold]]s. This difference in behavior is due ultimately to the fact that spacetimes have isotropy subgroups which are subgroups of the [[Lorentz group]] SO⁺(31,'''R'''3), which is a ''noncompact'' [[Lie group]], while four-dimensional Riemannian manifolds (i.e., with [[definite bilinear form|positive definite]] [[metric tensor]]), have isotropy groups which are subgroups of the [[compact group|compact]] Lie group SO(4).

Cartan showed that ''at most ten covariant derivatives are needed to compare any two Lorentzian manifolds'' by his method, but experience shows that far fewer often suffice, and later researchers have lowered his upper bound considerably. It is now known, for example, that

In 4 dimensions, Karlhede's improvement to Cartan's program reduces the maximal number of covariant derivatives of the Riemann tensor needed to compare metrics to 7. In the worst case, this requires 3156 independent tensor components.<ref>{{citation|first1=M. A. H.|last1=MacCallum|first2=J. E.|last2=Åman|title=Algebraically independent nth derivatives of the Riemannian curvature spinor in a general spacetime|journal=Classical and Quantum Gravity|volume=3|page=1133|year=1986|issue=6 |doi=10.1088/0264-9381/3/6/013|bibcode = 1986CQGra...3.1133M |s2cid=250892608 }}</ref> There are known models of spacetime requiring all 7 covariant derivatives.<ref>{{citation|first1=Robert|last1=Milson|first2=Nicos|last2=Pelavas|title=The type N Karlhede bound is sharp|journal=Class. Quantum Grav.|volume=25|year=2008|page=012001 |doi=10.1088/0264-9381/25/1/012001|arxiv=0710.0688|s2cid=15859985 }}</ref> For certain special families of spacetime models, however, far fewer often suffice. It is now known, for example, that

*at most one differentiation is required to compare any two [[null dust solution]]s.,

*at most two differentiations are required to compare any two Petrov '''D''' [[vacuum solution (general relativity)|vacuum solution]]s,

*at most three differentiations are required to compare any two perfect [[fluid solution]]s.<ref>{{cite book |author1=Stephani, Hans |author2=Kramer, Dietrich |author3=MacCallum, Malcolm |author4=Hoenselaers, Cornelius |author5=Hertl, Eduard | title=Exact Solutions to Einstein's Field Equations (2nd ed.) | ___location=Cambridge | publisher=Cambridge University Press | year=2003 | isbn=0-521-46136-7}}</ref>

*at most one differentiation is required to compare any two [[null dust solution]]s.

An important unsolved problem is to better predict how many differentiations are really necessary for spacetimes having various properties. For example, somewhere two and five differentiations, at most, are required to compare any two Petrov '''III''' vacuum solutions. Overall, it seems to safe to say that at most six differentiations are required to compare any two spacetime models likely to arise in general relativity.

Faster implementations of the method running under a modern symbolic computation system available for modern [[operating system]]s in common use, such as [[Linux]], would also be highly desirable. It has been suggested that the power of this algorithm has not yet been realized, due to insufficient effort to take advantage of recent improvements in [[differential algebra]]. The appearance in the "near future" of a proper on-line database of known solutions has been rumored for decades, but this has not yet come to pass. This is particularly regrettable since it seems very likely that a powerful and convenient database is well within the capability of modern software.

==See also==

*[[Vanishing scalar invariant spacetime]]

*[[Computer algebra system]]

*[[Frame fields in general relativity]]

*[[Petrov classification]]

==External links==

*[https://web.archive.org/web/20051023160408/http://130.15.26.66/servlet/GRDB2.GRDBServlet Interactive Geometric Database] includes some data derived from an implementation of the Cartan-KarlhedeCartan–Karlhede algorithm.

*[http://130.15.26.66/servlet/GRDB2.GRDBServlet Interactive Geometric Database] includes some data derived from an implementation of the Cartan-Karlhede algorithm.

==References==

<references/>

{{DEFAULTSORT:Cartan-Karlhede algorithm}}

*{{cite book | author=Stephani, Hans; Kramer, Dietrich; MacCallum, Malcom; Hoenselaers, Cornelius; Hertl, Eduard| title=Exact Solutions to Einstein's Field Equations (2nd ed.) | ___location=Cambridge | publisher=Cambridge University Press | year=2003 | isbn=0-521-46136-7}} Chapter 9 offers an excellent overview of the basic idea of the Cartan method and contains a useful table of upper bounds, more extensive than the one above.

*{{cite journal | author=Pollney, D.; Skea, J. F.; and d'Inverno, Ray | title=Classifying geometries in general relativity (three parts) | journal=Class. Quant. Grav. | year=2000 | volume=17 | pages=643–663, 2267–2280, 2885–2902 | doi=10.1088/0264-9381/17/3/306}} A research paper describing the authors' database holding classifications of exact solutions up to local isometry.

*{{cite book | author=Olver, Peter J. | title=Equivalents, Invariants, and Symmetry | ___location=Cambridge | publisher=Cambridge University Press | year=1995 | isbn=0-521-47811-1}} An introduction to the Cartan method, which has wide applications far beyond general relativity or even Riemannian geometry.

[[Category:Riemannian geometry]]

[[Category:Mathematical methods in general relativity]]